Video: Calculating Probability in a Real Life Context Involving Areas

Alice is randomly tossing shoes onto her rug which is modelled in the diagram. What is the probability that the shoes will land in the shaded area of the rug? Give your answer as a fraction in its simplest form.

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Video Transcript

Alice is randomly tossing shoes onto her rug which is modelled in the diagram. What is the probability that the shoes will land in the shaded area of the rug? Give your answer as a fraction in its simplest form.

So in the diagram, what we can see is we’ve got a rug, and the rug is eight foot by 16 foot. And within the rug, there’s a shaded area which is four foot by 10 foot. So now, in order to work out what probability that the shoe will land in the shaded area is, we need to work out what fraction of the rug is shaded. And to do that, we’re gonna have to work out the area of the rug and the area of the shaded section. And to do that, we’ve got a formula. And that formula is that the area of a rectangle is equal to the length multiplied by the width. So what I’m gonna do is call the total area of the rug 𝐴 and the area of the shaded region 𝐵.

So, first of all, to work out the area of the total rug, so the area of 𝐴, it’s gonna be equal to 16 multiplied by eight. And that’s 16 foot multiplied by eight foot which is gonna be equal to 128 square feet. And we could work that out mentally because we could do 10 multiplied by eight which is 80. Six multiplied by eight which is 48. Add them together, gives us our 128. Okay, great, so we’ve got the total area of the rug. Now, we need to move on and find the area of the shaded region. So the area of the shaded region, region 𝐵, is equal to 10 multiplied by four. That’s 10 foot multiplied by four foot which is gonna be equal to 40 square feet.

So now, we’ve got the areas of both the total rug and the shaded region. What we want to do is work out what fraction of the rug is shaded. Because that’s gonna give us the probability that the shoe will land in the shaded area of the rug. So, therefore, we can say that the probability that the shoe is gonna land in the shaded area is gonna be equal to 40 divided by 128 or 40 over 128. And that’s because 40 was the area of the shaded region. And 128 is the total area of the rug. And we said that the fraction of the rug that is shaded is gonna be the same as the probability that the shoes will land in the shaded area. So have we finished? Have we solved the problem?

Well, if we take a look at the question, it says that we want the answer as a fraction which we’ve got. So yes, we’ve done that. However, it’s not in its simplest form. So we haven’t quite finished. We need to carry out another stage. Well, in order to simplify the fraction, what we wanna do is find a factor that goes into both 40 and 128. And what we’re really looking for is the highest common factor. We can see that the highest common factor of 40 and 128 is eight. And therefore, if we divide 40 by eight, we get five. And we divide 128 by eight, we get 16.

Now, you might think, “Well, how would I know that eight goes into both of these numbers?” Well, 40, we know that eight goes into five times. And that’s quite straightforward cause we should know that from our times tables. And if we refer back to the early part of the question, we can see that we already worked out 16 multiplied by eight, was 128. So therefore, we know that eight goes into 128 16 times. So, therefore, we can say that the probability that the shoes will land in the shaded area of the rug, as a fraction in its simplest form, is equal to five over 16 or five-sixteenths.

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